Transmission through Cantor structured Dirac comb potential

TL;DR

This paper introduces a novel fractal-based quantum potential system called the Cantor Dirac comb, analyzes its transmission properties, and reveals unique features like band-like transmission and sharp resonances, expanding understanding of quantum tunneling in fractal potentials.

Contribution

It is the first study to analyze quantum tunneling through a fractal geometric Dirac comb potential, deriving a closed-form transmission expression and exploring its unique transmission features.

Findings

Identification of band-like transmission features.

Scaling behavior of reflection coefficient with wave vector.

Presence of sharp transmission resonances.

Abstract

In this study, we introduce the Cantor-structured Dirac comb potential, referred to as the Cantor Dirac comb (CDC-$\rho_{N}$) potential system, and investigate non-relativistic quantum tunneling through this novel potential configuration. This system is engineered by positioning delta potentials at the boundaries of each rectangular potential segment of Cantor potential. This study is the first to investigate quantum tunneling through a fractal geometric Dirac comb potential. This potential system exemplifies a particular instance of the super periodic potential (SPP), a broader class of potentials that generalize locally periodic potentials. Utilizing the theoretical framework of SPP, we derived a closed-form expression for the transmission probability for this potential architecture. We report various transmission characteristics, including the appearance of band-like features and the scaling behavior of the reflection coefficient with wave vector $k$, which is governed by a scaling function expressed as a finite product of the Laue function. A particularly striking feature of the system is the occurrence of sharp transmission resonances, which may prove useful in applications such as highly sharp transmission filters.